Second, although this did not appear in our illustrations in Section , an especially challenging aspect of the dynamic treatment regime problem is non-regularity, resulting from the non-smooth “max” operator in (REF ), that affects the limit distribution theory of the M/Z-estimators and, in turn, the corresponding bootstrap-based inference. A common remedy for failure of the bootstrap, e.g., due to non-regularity, is to “under-sample” with the so-called m-out-of-n bootstrap. That is, [1]} showed that, by taking bootstrap samples of size \(m=o(n)\) , bootstrap failure could be avoided. A more sophisticated \(m\) -out-of-\(n\)  bootstrap scheme was proposed in [2]} that could prevent bootstrap failure in the dynamic treatment regime setting resulting from the non-regularity induced by the “max” operator. An interesting follow-up project could investigate the reliability of the proposed generalized IM equipped with a \(m\) -out-of-\(n\)  bootstrap strategy under non-regularity.
